Analytic estimate of thermal noise in sphere optical reference cavities

Thermal noise in optical cavities sets a fundamental limit to the frequency instability of ultra-stable lasers. We estimate the contribution of the sphere spacer using the new Equation based on Levin's formulation of the fluctuation-dissipation-theorem (FDT) and elastic strain energy. Focusing on the sphere spacer geometry, the estimate and simulation of the thermal noise have been carried out. We find that thermal noise of the sphere spacer behaves different from cylindrical cavities. The estimate for the sphere spacer by the new Equation agrees better with simulation compared with previous equations.